Earth's_internal_heat_budget

# Earth's internal heat budget **Earth's internal heat budget** is the accounting of the [[Heat|heat]] leaving the planet's interior, of the sources that supply it, and of the store that is being drawn down. It is a small quantity beside the solar term of [[Earth's_energy_budget|Earth's energy budget]] — smaller by more than three orders of magnitude — and it is nevertheless the budget that drives every process the Sun cannot reach: the slow overturn of the mantle, the drift of plates, volcanism, and the long-lived magnetic field. Two sources feed it. One is primordial: the [[Energy|energy]] released by accretion and by the sinking of iron to the core, still leaking out of a body whose [[Thermal_diffusivity|thermal diffusivity]] makes it a very poor conductor for its size. The other is radiogenic, the continuing [[Radioactive_decay|decay]] of four long-lived [[Isotope|isotopes]] distributed through the silicate Earth. In the microsim below the reader scrubs a single control — time, from 4.57 Ga to the far future — and watches four bars fall at four different rates. The governing expression is `P(t) = sum_i P_i(0) 2^(-t/T_i)`, with half-lives of 1.3 Gyr for ⁴⁰K, 14.0 Gyr for ²³²Th, 0.704 Gyr for ²³⁵U and 4.47 Gyr for ²³⁸U.[^earle-40k][^murphy-u][^murphy-th] Two readouts hang off the same slider: the ²³⁵U/²³⁸U ratio, which stands at about 140:1 today and was 1:1 roughly six billion years ago, and, as a caption, the age Lord Kelvin obtained by treating the whole planet as a cooling solid with no sources at all.[^murphy-u][^thomson1864] On the [[Energy]] flagship this article is the child of Part IV — Scientific use, the section on Earth's internal heat, where it places [[Thorium|thorium]], [[Uranium|uranium]] and [[Potassium|potassium]] as the planet's own fuel. It is the point where the [[Half-life|half-life]] law of Part III stops being a laboratory curve and becomes a geological clock running at four speeds. ## Heat and early estimate of Earth's age The first quantitative age of the Earth came from its heat, and it was wrong for a reason that turned out to be the subject of this article. William Thomson, later Lord Kelvin, took the planet to have started molten, to have cooled by [[Thermal_conduction|conduction]] alone, and to have no internal sources; the measured increase of [[Temperature|temperature]] with depth then fixes how long the cooling has run. In "On the secular cooling of the Earth" he put the answer between 20 and 400 million years, and over the following decades he narrowed it toward the low end.[^thomson1864] Geologists and biologists objected that their own evidence needed far longer, and John Perry showed in 1895 that a mantle able to move heat by [[Convection|convection]] rather than conduction would stretch the estimate by an order of magnitude or more.[^perry1895] Radioactivity settled it twice over. It supplied the missing source term, so that the [[Geothermal_gradient|geothermal gradient]] is not simply the fading memory of a hot start; and it supplied an independent clock. The decay law itself is the whole of that clock: after n half-lives a population is reduced to N = N₀·(1/2)ⁿ, and the survival probability to a time t is (1/2)^(t/T½).[^murphy-halflife] Each individual atom's lifetime is random; the population's is exact, which is why a rock can be dated and a single nucleus cannot. Steven Earle states the rule for the isotope that matters most here: half of any ⁴⁰K decays in 1.3 billion years, half of the remainder in the next 1.3 billion, and three half-lives — 3.9 Ga — leave 12.5 % of the original.[^earle-40k] The usable systems differ enormously in reach. The tabulated half-lives and working ranges are 1.3 Ga for K-Ar, useful from 10 ka to the age of the Earth; 4.5 Ga for U-Pb over 1 Ma to 4.57 Ga; 47 Ga for Rb-Sr, which needs 10 Ma before it says anything; and 5,730 years for ¹⁴C, which is exhausted by about 60 ka.[^earle-table82] Rubidium's clock is slow enough that only about 6.5 % of it has decayed in the whole history of the planet (computed), which is why Rb-Sr ages are imprecise. Every one of these methods assumes a closed system with no daughter product at formation, an assumption checked rather than presumed.[^earle-40k] ## Global internal heat flow The flux out of the solid Earth is measured as a temperature gradient in boreholes multiplied by the [[Thermal_conductivity_and_resistivity|thermal conductivity]] of the rock, and it is small: a few tens of milliwatts per square metre, against the 236 W/m² of sunlight the planet absorbs. Summed over the surface it is of order forty terawatts, and roughly half of that is usually attributed to the decay of the four isotopes treated below, the remainder to the cooling of a hot interior.[^heatflow-external] The flux is not uniform. It is highest over young oceanic crust and lowest over old continental shields, for a reason that is pure conduction: a slab of hot rock emplaced at a ridge and then left to cool from its top surface loses heat at a rate that falls as the inverse square root of its age, so the heat-flow map of the ocean floor is essentially an age map. Continents behave differently because their [[Crust_(geology)|crust]] is thick, old and enriched in exactly the elements that supply radiogenic heat, so a large share of a continental measurement is made in the crust itself rather than conducted up from below. The order-of-magnitude comparison with the surface budget is worth stating plainly. The internal flux is smaller than the noon-to-midnight swing in net radiation over a field in Iowa, smaller than the error bar on most terms of the solar budget, and completely irrelevant to the planet's surface [[Temperature|temperature]]. What it is not irrelevant to is everything below the surface, where it is the only energy source there is. [[Geothermal_energy|Geothermal]] extraction lives on the contrast: the flux is feeble, but the store it has built up over billions of years is not. ## Sources of heat Two categories, with a blurred boundary. Primordial heat is the energy delivered once, at the beginning, and still in transit; radiogenic heat is energy delivered continuously since. The boundary blurs because much of the primordial store was itself topped up by short-lived radioactivity in the first few million years, and because the core's ongoing crystallisation releases latent heat that is neither. The microsim treats only the radiogenic term, because that is the term whose time dependence can be written down exactly. ### Radiogenic heat Four nuclides carry essentially all of the present radiogenic budget: ⁴⁰K, ²³²Th, ²³⁵U and ²³⁸U. Each contributes P_i(0)·2^(−t/T_i), and because the half-lives span a factor of twenty the mix changes completely over geological time. Running the exponentials back to 4.57 Ga from today's abundances (derived from the cited half-lives) gives the factors by which each was more plentiful at formation: ⁴⁰K by 11.4, ²³²Th by 1.25, ²³⁵U by 90, and ²³⁸U by 2.03. Potassium and ²³⁵U therefore dominated the young Earth and have since faded; thorium, with a half-life three times the age of the planet, has barely changed and will still be running when the others are gone. That is the shape the reader sees in the bars, and it means the radiogenic contribution to the budget has fallen by a factor of several since the Archean — a fact that constrains how vigorously the mantle could have convected then. The uranium ratio is the sim's sharpest readout because it is directly observable. Natural [[Uranium|uranium]] today is 0.72 % ²³⁵U against 99.2745 % ²³⁸U, a ratio of about 140:1; with half-lives of 0.704 and 4.47 Gyr the two were equally abundant about six billion years ago, and solving 140 = 2^(t(1/0.704 − 1/4.47)) gives 5.96 Gyr (computed).[^murphy-u] Running the same arithmetic forward to the planet's formation gives about 24 % ²³⁵U at 4.57 Ga (derived) — uranium ore that would have been reactor-grade without enrichment, which is why natural fission reactors were possible in the Precambrian and are not now. ILLUSTRATIVE: the sim takes today's apportionment of the ~20 TW radiogenic term among the four isotopes as an external input and propagates only the decay factors, so its bar heights are exact in their ratios through time and only as good as that input in absolute terms.[^heatflow-external] Where these elements sit matters as much as how much there is. Potassium, [[Thorium|thorium]] and uranium are incompatible in the mineralogical sense: they do not fit comfortably into the crystal lattices of mantle minerals, so partial melting concentrates them upward. Over 4.5 billion years that has loaded the continental crust with a disproportionate share of the planet's radiogenic inventory, which is why a granite terrain is measurably warmer at depth than an ocean basin of the same age. ### Primordial heat The other half of the budget is left over. Accretion converted the gravitational energy of infalling material into heat; the separation of iron from silicate released more as the dense phase sank to the centre; and giant impacts, including the one usually invoked to form the Moon, delivered enough energy to melt a substantial fraction of the planet. None of this had anywhere to go quickly. Rock conducts heat badly, and a sphere 6,371 km in radius has a conductive time constant far longer than the age of the solar system, so the interior is still losing a store laid down at the start. Convection in the mantle short-circuits the conduction problem for the bulk of the planet, but the boundary layers at top and bottom remain conductive, and they set the rate. A separate primordial term is still being released today: as the liquid outer core freezes onto the solid inner core, it gives up latent heat and expels light elements whose buoyancy drives the compositional convection that maintains the geodynamo. That coupling is why the internal budget is not merely a thermal question — the magnetic field is powered by it. ## Heat flow and tectonic plates Plate motion is the surface expression of mantle convection, and convection is how a planet with a poor [[Thermal_conductivity_and_resistivity|conductivity]] gets its heat out anyway. A plate is the cold upper thermal boundary layer of the convecting system: it forms at a ridge, thickens and cools as it ages, becomes denser than the material beneath it, and eventually sinks. The sinking is not a consequence of the motion but the cause of most of it, which is why the fastest plates are the ones with the most subducted edge. The timescales are the geologists' own. Continental drift proceeds at about 10⁻⁸ km per hour — some 9 cm per year (derived) — and the magnetic record calibrates it: the present normal-polarity interval has lasted 780,000 years, and crust formed between 780 and 900 ka ago carries reversed magnetisation, so the width of those stripes divided by their duration is a spreading rate.[^earle-magnetic] The Juan de Fuca plate shows how short an ocean plate's life can be: its oldest unsubducted crust is just over 8 million years old, and the part now entering the trench is between 0 and 6 million years old.[^earle-jdf] Set against the 4.57-billion-year clock those numbers are vanishingly recent. Compressing the planet's history into a single year makes one day 12.5 million years and one hour about half a million; the whole Phanerozoic, with every animal that has ever left a fossil, occupies the last six weeks, and the last glaciation ended 81 seconds before midnight.[^earle-year] The plates have rearranged the surface many times over inside that year, and the energy for all of it came from the budget described above — which, per unit area, is smaller than the heat a human body gives off. ## See also - [[Radiogenic_nuclide]] - [[Geothermal_gradient]] - [[Earth's_energy_budget]] - [[Half-life]] - [[Decay_heat]] - [[Decay_chain]] - [[Geothermal_energy]] ## External links - [Physical Geology](https://open.umn.edu/opentextbooks/textbooks/physical-geology), Earle (2015) — Portal Book 128, Chapter 8 for the dating systems and the geological year, Chapter 9 for Earth's interior - [Energy and Human Ambitions on a Finite Planet](https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet), Murphy (2021) — Portal Book 097, Chapter 15 for the decay law and the uranium isotopes - The Wikipedia pair's external links list the heat-flow compilations and geoneutrino results that would pin the present radiogenic power ## References [^thomson1864]: Thomson, William (1864). "On the secular cooling of the Earth." *Transactions of the Royal Society of Edinburgh* 23: 157–169. (Read 1862; the source of the 20–400 million-year estimate later narrowed toward its lower bound.) [^perry1895]: Perry, John (1895). "On the age of the Earth." *Nature* 51: 224–227, with further exchanges in the same volume. (The convective objection to Thomson's conductive model.) [^murphy-halflife]: Murphy, Thomas W. (2021). *Energy and Human Ambitions on a Finite Planet*. Chapter 15 (nuclear energy), pp. 262–265 (N = N₀(1/2)ⁿ and the survival probability (1/2)^(t/T½); Table 15.1's 16-million-neutron column; Table 15.2's decay modes; the free neutron at 10.25 min and ³H at 12.32 yr). Portal Book 097, https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet [^murphy-u]: Murphy (2021), *Energy and Human Ambitions on a Finite Planet*, Chapter 15, p. 277 (natural uranium is 0.72 % ²³⁵U and 99.2745 % ²³⁸U, about 140:1; half-lives 0.704 Gyr and 4.47 Gyr; the two were equally abundant about 6 Gyr ago). Portal Book 097. The 5.96 Gyr and 24 %-at-4.57-Ga figures quoted here are computed from those inputs, the first of them also in sub-manual 10 §4.1. [^murphy-th]: Murphy (2021), *Energy and Human Ambitions on a Finite Planet*, Chapter 15, p. 279 (²³²Th half-life 14.0 Gyr; also the ²³⁹U → ²³⁹Np → ²³⁹Pu breeder chain). Portal Book 097. [^earle-40k]: Earle, Steven (2015). *Physical Geology*. Chapter 8 "Measuring Geological Time" (pp. 222–249), p. 237 (half of any ⁴⁰K decays in 1.3 Ga and half the rest in the next 1.3 Ga; three half-lives leave 12.5 %; isotopic ages require a closed system with no daughter at formation). Portal Book 128, https://open.umn.edu/opentextbooks/textbooks/physical-geology [^earle-table82]: Earle (2015), *Physical Geology*, Chapter 8, p. 239, Table 8.2 (half-life and useful range: K-Ar 1.3 Ga, 10 ka–4.57 Ga; U-Pb 4.5 Ga, 1 Ma–4.57 Ga; Rb-Sr 47 Ga, 10 Ma–4.57 Ga; ¹⁴C 5,730 y, 100 y–60 ka). Portal Book 128. The 6.5 % figure for rubidium decayed over Earth's history is computed in sub-manual 10 §4.1, not printed in the book. [^earle-magnetic]: Earle (2015), *Physical Geology*, Chapter 8, pp. 242–243 (the Brunhes normal chron has lasted 780,000 years; crust formed 780–900 ka ago is reversed; the chronology is reliable back to about 250 Ma). Portal Book 128. [^earle-jdf]: Earle (2015), *Physical Geology*, Chapter 8, p. 244 (the oldest unsubducted Juan de Fuca crust is just over 8 Myr old and the subducting part 0–6 Myr; magnetic polarity alone cannot identify which normal interval a rock records). Portal Book 128. [^earle-year]: Earle (2015), *Physical Geology*, Chapter 8, pp. 246–247 (the geological year: one day is ≈ 12.5 Myr and one hour ≈ 500,000 yr; Earth forms on January 1 at 4.57 Ga; the last glaciation ends 81 seconds before midnight; continental drift ≈ 10⁻⁸ km/h). Portal Book 128. Sub-manual 10 §4.4 records an arithmetic slip in the book's Exercise 8.5 and gives 12.52 Myr/day as the exact scale factor; the 9 cm/yr drift rate quoted here is the book's 10⁻⁸ km/h converted (derived). [^heatflow-external]: The present global heat flow (of order 40 TW) and the ~20 TW radiogenic share are external to sub-manuals 04, 05 and 10, which extract no heat-flow figure. Portal Book 128 covers the interior at Chapter 9 "Earth's Interior", pp. 250–271, and Chapter 10 "Plate Tectonics", pp. 272–308 (pages to pin), and the apportionment of today's radiogenic power among ⁴⁰K, ²³²Th, ²³⁵U and ²³⁸U, which the sim takes as its input, is likewise a page to pin. A borehole heat-flow compilation or a geoneutrino measurement would settle the split.  **Microsim — three.js (Wikitube framework), pending deploy:** *Earth's internal heat budget* will play here once `https://wikitube-3d-microsims.netlify.app/matter/Earth's_internal_heat_budget.html` is live.   ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Earth's_internal_heat_budget) : [Wikitube](https://en.wikitube.io/wiki/Earth's_internal_heat_budget) · pinned revision [1370077793](https://en.wikipedia.org/w/index.php?oldid=1370077793) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Energy]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Energy row E27 · sim pending (matter/Earth's_internal_heat_budget).*